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Fundamental Forms


There are three types of so-called fundamental forms. The most important are the first and second (since the third can be expressed in terms of these). The fundamental forms are extremely important and useful in determining the metric properties of a surface, such as line element, area element, normal curvature, Gaussian curvature, and mean curvature. Let M be a regular surface with v_(p),w_(p) points in the tangent space M_(p) of M. Then the first fundamental form is the inner product of tangent vectors,

 I(v_(p),w_(p))=v_(p)·w_(p).
(1)

For M in R^3, the second fundamental form is the symmetric bilinear form on the tangent space M_(p),

 II(v_(p),w_(p))=S(v_(p))·w_(p),
(2)

where S is the shape operator. The third fundamental form is given by

 III(v_(p),w_(p))=S(v_(p))·S(w_(p)).
(3)

The first and second fundamental forms satisfy

I(ax_u+bx_v,ax_u+bx_v)=Ea^2+2Fab+Gb^2
(4)
II(ax_u+bx_v,ax_u+bx_v)=ea^2+2fab+gb^2,
(5)

where x:U->R^3 is a regular patch and x_u and x_v are the partial derivatives of x with respect to parameters u and v, respectively. Their ratio is simply the normal curvature

 kappa(v_(p))=(II(v_(p)))/(I(v_(p)))
(6)

for any nonzero tangent vector. The third fundamental form is given in terms of the first and second forms by

 III-2HII+KI=0,
(7)

where H is the mean curvature and K is the Gaussian curvature.

The first fundamental form (or line element) is given explicitly by the Riemannian metric

 ds^2=Edu^2+2Fdudv+Gdv^2.
(8)

It determines the arc length of a curve on a surface. The coefficients are given by

E=||x_u||^2=|(partialx)/(partialu)|^2
(9)
F=x_u·x_v=(partialx)/(partialu)·(partialx)/(partialv)
(10)
G=||x_v||^2=|(partialx)/(partialv)|^2.
(11)

The coefficients are also denoted g_(uu)=E, g_(uv)=F, and g_(vv)=G. In curvilinear coordinates (where F=0), the quantities

h_u=sqrt(g_(uu))=sqrt(E)
(12)
h_v=sqrt(g_(vv))=sqrt(G)
(13)

are called scale factors.

The second fundamental form is given explicitly by

 edu^2+2fdudv+gdv^2
(14)

where

e=sum_(i)X_i(partial^2x_i)/(partialu^2)
(15)
f=sum_(i)X_i(partial^2x_i)/(partialupartialv)
(16)
g=sum_(i)X_i(partial^2x_i)/(partialv^2),
(17)

and X_i are the direction cosines of the surface normal. The second fundamental form can also be written

e=-N_u·x_u
(18)
=N·x_(uu)
(19)
f=-N_v·x_u
(20)
=N·x_(uv)
(21)
=N·x_(vu)
(22)
=-N_u·x_v
(23)
g=-N_v·x_v
(24)
=N·x_(vv),
(25)

where N is the normal vector (Gray 1997, p. 368), or

e=(det(x_(uu)x_ux_v))/(sqrt(EG-F^2))
(26)
f=(det(x_(uv)x_ux_v))/(sqrt(EG-F^2))
(27)
g=(det(x_(vv)x_ux_v))/(sqrt(EG-F^2))
(28)

(Gray 1997, p. 379).


See also

Arc Length, Area Element, First Fundamental Form, Gaussian Curvature, Geodesic, Kähler Manifold, Line of Curvature, Line Element, Mean Curvature, Normal Curvature, Riemannian Metric, Scale Factor, Second Fundamental Form, Surface Area, Third Fundamental Form, Weingarten Equations

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References

Gray, A. "The Three Fundamental Forms." §16.6 in Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 368-371 and 380-382, 1997.

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Fundamental Forms

Cite this as:

Weisstein, Eric W. "Fundamental Forms." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/FundamentalForms.html

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